{"id":"21f3c3e0-ae35-48f7-8c9f-29649c81f6b8","arxiv_id":"1908.04560","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Steane enlargement of Cartesian-product codes is proven to add at least K quantum dimensions, with an algorithm giving the exact gain, and several resulting codes meet or beat the Gilbert-Varshamov bound.","lead":"This paper constructs new quantum error-correcting codes by applying Steane enlargement, a known dimension-increasing technique, to codes built from Cartesian product point sets. It gives a lower bound on the dimension gain and an algorithm to compute the exact gain, with several resulting codes matching or beating the Gilbert-Varshamov bound.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Tables 1–4 require C(L(δ)) to contain its dual (C(L(δ))^⊥ ⊆ C(L(δ))), a property the paper cites from [4] but never proves; if it fails for any row, the listed Steane-enlarged code does not exist.","rationale":"The central claim is that Steane enlargement of C(L(δ)) yields quantum codes with the listed parameters. The proof of Proposition 5 is sound given the dual-containment hypothesis, and the dimension/distance calculations check out. The load-bearing unresolved point is precisely whether that hypothesis holds for the examples: the tables in Section 3.2 rely on parameters and self-orthogonality taken from [4], without reproducing or precisely citing the part of [4] that supplies it. This is not an internal inconsistency of Proposition 5, but it is a genuine gap in the paper's argument for its headline examples, because the condition is known not to be automatic. The reader's weakest_assumption identifies exactly this point, and I agree. The additional typos in Algorithm 1 and Proposition 10 are concrete and worth fixing, but they do not affect the existence of the codes with the lower-bound dimensions; they only affect the correctness of the printed algorithm for computing the exact increase. Therefore the appropriate verdict remains CONDITIONAL, unchanged from the reader.","tokens_in":9788,"tokens_out":39698,"duration_ms":348328,"concrete_test":"Write a short script that, for each row in Tables 1–4, enumerates all exponent vectors a∈∏_{i=1}^m{0,...,p^{r_i}-1} with ∏(a_i+1)<δ and verifies ∏(p^{r_i}-a_i)≥δ. Since n≤1024, this exhaustive check is exact for the paper's examples. If every row passes, the external dual-containment assumption is valid for the tables; if any row fails, that quantum code does not exist and must be removed from the paper. An alternative settlement is to identify the exact proposition in [4] that proves L^⊥(δ)⊆L(δ) for these parameters.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's concrete payoff is the codes in Tables 1–4, obtained by applying Proposition 5 to C=C(L(δ)). Proposition 5 is explicitly conditional on C being 'self-orthogonal' in the sense that it contains its Euclidean dual, C^⊥⊆C. The paper never proves this condition for the δ and r values in the tables; it inherits the parameters and the dual-containment from Galindo et al. [4] without stating which theorem there establishes it. The condition is not automatic. By Proposition 4, C(L(δ))^⊥=C(L^⊥(δ)), so C(L(δ))^⊥⊆C(L(δ)) is equivalent to L^⊥(δ)⊆L(δ), i.e., every exponent vector a with ∏(a_i+1)=μ(a)<δ must satisfy σ(a)=∏(p^{r_i}-a_i)≥δ. This can fail: for r=(1,1), q=2, δ=3, a=(1,0) has μ=2<3 but σ=2<3, so C(L(3)) does not contain its dual. The same mechanism could fail for a row of the tables even though the quoted parameters match. If any row fails this inclusion, that row's CSS code and its Steane enlargement are invalid, and the corresponding GV-bound comparison collapses. The paper should either prove L^⊥(δ)⊆L(δ) for the table parameters, supply a precise citation to a stated theorem in [4], or explicitly mark the examples as conditional on that external verification. The Algorithm 1 base-case typo and the Proposition 10 equality error are real but secondary; they affect reproducibility of the 'actual increase' column, not the existence of the lower-bound codes.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Steane enlargement of the Cartesian-product codes C(L(δ)) previously considered by Galindo et al. The authors observe that C(L(δ−1)) contains C(L(δ)) and gains at least K monomials, where K is the largest index with δ−1 ≤ p^{r_K}. Proposition 5 converts this into a quantum code [[n, ≥2k−n+K, ≥δ]]_q whenever C(L(δ)) contains its Euclidean dual and δ ≤ p^{r_2}+1. Section 3.1 defines τ^{(r)}(s), the number of exponent tuples with σ-value s, and uses it to compute the exact dimension increase; Proposition 7 and Corollary 9 give lower bounds and characterize when Proposition 5 is sharp. Algorithm 1 is proposed for computing τ. The final section gives tables of enlarged codes for q = 5, 8, 9, 16, compares them with the Gilbert–Varshamov and Singleton bounds, and identifies two MDS codes.","tokens_in":10181,"tokens_out":14269,"duration_ms":128605,"significance":"Conditional on the dual-containment hypothesis, the main inequality is correct: the relative-distance reduction d(C′,C′⊥) = d(C′) = δ−1 is justified by the nesting in Theorem 2, and the ceiling identity holds for δ−1 ≤ q. The paper gives a useful exact-counting method for the dimension increase and supplies concrete tables that exhibit codes at or above the Gilbert–Varshamov bound, including two Singleton-optimal codes over smaller fields than earlier constructions. These are concrete, checkable contributions. However, the tables’ validity depends on a self-orthogonality property that is asserted rather than proved or precisely cited, and the printed Algorithm 1 and Proposition 10 contain errors that affect reproducibility. The central idea is sound and the flaws are local, so major revision is appropriate.","major_comments":[{"comment":"The tables and the numerical examples apply Proposition 5 to C(L(δ)) under the assumption that C(L(δ)) contains its Euclidean dual, but this dual-containment is never proved in the manuscript. By Proposition 4, C(L(δ))⊥ = C(L⊥(δ)), so the required inclusion is equivalent to L⊥(δ) ⊆ L(δ), i.e. μ(a) < δ ⇒ σ(a) ≥ δ for all a ∈ Δ(r). This condition is not automatic: for q = 2, r = (1,1), δ = 3, the monomial a = (1,0) satisfies μ(a) = 2 < 3 but σ(a) = 2 < 3, so L⊥(3) ⊄ L(3) and C(L(3)) does not contain its dual. Since the first column of each table and the subsequent enlargement both depend on this property, the paper must either prove L⊥(δ) ⊆ L(δ) for every row, cite the exact statement in [4] or [13] that establishes it, or explicitly mark the examples as conditional on an external verification. As written, a single failing row would invalidate that row's enlarged code and its comparison with the Gilbert–Varshamov bound.","section":"Section 3.2, Tables 1–4; Proposition 5"},{"comment":"Proposition 10 states that Algorithm 1 computes τ^{(r)}(δ) = K, but this is false as written. Algorithm 1 computes τ^{(r)}(s) for arbitrary s; the quantity relevant to Steane enlargement is τ^{(r)}(δ−1), not τ^{(r)}(δ), and it is not generally equal to K. For example, the δ = 5 row of Table 1 has K = 2 but actual increase 3, so τ^{(r)}(4) = 3 ≠ K. The statement should say that Algorithm 1 computes the exact dimension increase τ^{(r)}(δ−1), and the column header 'Prop. 10' must be interpreted accordingly. In addition, the base case of Algorithm 1 says 'return 1 if s ≤ r1' but should be 's ≤ p^{r1}'; otherwise the algorithm gives wrong answers whenever p^{r1} > s > r1. These are not merely cosmetic because the 'actual increase' column is the paper's main computational output.","section":"Section 3.1, Proposition 10 and Algorithm 1"}],"minor_comments":[{"comment":"The sentence 'we may also split s in two factors s = f1f2 such that f1f2 < s ≤ p^{r_K}' is impossible; it should state that s has a nontrivial factorization s = f1f2 with 1 < f1, f2 < s, with f1 = f2 possible when s is a square. The intended counting is clear, but the printed argument needs correction.","section":"Section 3.1, proof of Proposition 7"},{"comment":"The text 'the Steane-enlargement described in Proposition 3' should refer to Proposition 5.","section":"Section 3, remark after Proposition 5"},{"comment":"The sentence 'Proposition 5 gave the true minimal distance' should read 'true dimension increase'; the minimal distance was not in question.","section":"Example 2"},{"comment":"The claim that the number of d's considered in Algorithm 1 is ∏_{i=1}^{m−1} p^{r_i} = n/p^{r_m} is inconsistent with the loop condition d|s; the actual number of leaves is at most τ^{(r)}(s), and the stated O(n/p^{r_m}) bound needs justification or correction. This does not affect the correctness of the returned value.","section":"Section 3.1, complexity statement before Proposition 10"}],"recommendation":"major_revision","confidential_remarks":"The central construction is sound, but the manuscript needs a correct statement of Proposition 10, a corrected Algorithm 1 base case, and a precise justification of the dual-containment hypothesis used in all tables. These are fixable within the manuscript's scope, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper takes the Cartesian-product codes from Galindo et al. and applies Steane enlargement to them. The main result, Proposition 5, is clean: if C(L(δ)) is self-orthogonal and δ ≤ p^{r2}+1, then enlarging via C(L(δ−1)) gives a quantum code with dimension gain K, where K is the number of coordinates with δ−1 ≤ p^{ri}. The proof is correct, and the observation that relative distances can be replaced by ordinary distances for these nested codes is a neat and valid step. The generalization of the τ map to compute the exact dimension gain, not just a lower bound, is a genuine addition, and the tables show several codes that meet or beat the Gilbert–Varshamov bound, two meeting the quantum Singleton bound. That is a real, if modest, contribution to the quantum code tables.\n\nThe counting arguments and the dual description in Proposition 4 are sound. The algorithm's idea is right, and the examples are illustrative.\n\nNow the soft spots. First, Algorithm 1 has a wrong base case: it says return 1 if s ≤ r1, but it should be s ≤ p^{r1}. Second, Proposition 10 states that Algorithm 1 computes τ^{(r)}(δ) = K, but the dimension gain is τ^{(r)}(δ−1) — the statement is off by one. These are typos, but they will trip up anyone trying to reproduce the tables. Third, and more substantive: the tables assume each C(L(δ)) is self-orthogonal, but the paper never proves this for the listed parameters nor cites the specific theorem in [4] that establishes it. The condition is not automatic — it fails for q=2, r=(1,1), δ=3. For the table parameters, the condition likely holds, but as written the examples rest on an unverified assumption. Since the tables are the concrete payoff, this gap should be closed before publication, either with a short lemma or a precise pointer to [4].\n\nThe citation pattern is fine. The self-citation to [2] is appropriate, since the τ map generalizes that earlier work. The comparison to [4] and [13] is honest, including noting where the older codes do better.\n\nWho is this for? People who care about the fine print of quantum code tables and about Steane-enlargement techniques. It does not resolve an open problem, but it is a useful, correct construction with a few rough edges. Send it to peer review; ask for the typos to be fixed and the self-orthogonality condition to be justified. If that condition fails for some table row, the paper needs revision, not rejection.","headline":"Steane enlargement for Cartesian-product codes: a solid, useful construction with fixable presentation bugs and one unproven self-orthogonality assumption in the tables.","tokens_in":10707,"tokens_out":6158,"would_cite":true,"duration_ms":53475,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B27","81Q99"],"pacs":[],"model":"deepseek-v4-flash","headline":"Steane enlargement of Cartesian-product codes is shown to raise quantum-code dimension by an exactly computable amount, with tables of codes matching or beating known bounds.","keywords":["Cartesian product","Quantum code","Steane-enlargement","Finite fields","CSS construction","self-orthogonal codes","Gilbert-Varshamov bound","monomial Cartesian codes"],"falsifier":"For any row of the tables, compute the Euclidean dual $C(L(\\delta))^\\perp$ and check whether it is contained in $C(L(\\delta))$; the claimed enlarged quantum code exists only if containment holds. Alternatively, enumerate all exponent tuples with $\\sigma$-value $\\delta-1$ and compare the count with the output of Algorithm 1.","tokens_in":9582,"feed_emoji":"🧮","tokens_out":8016,"duration_ms":71364,"temperature":0.7,"pith_summary":"This paper establishes how much Steane enlargement can increase the dimension of quantum error-correcting codes built from classical codes on Cartesian product point sets. For any self-orthogonal code $C(L(\\delta))$ in this family with $2<\\delta\\le p^{r_2}+1$, enlarging by the nested code $C(L(\\delta-1))$ produces a quantum code with parameters $[[n, \\ge 2k-n+K, \\ge \\delta]]_q$, where $n$ is the code length, $k$ the classical dimension, and $K$ the number of coordinate directions whose field size is at least $\\delta-1$. The exact dimension gain is shown to be $\\tau^{(r)}(\\delta-1)$, the number of ways to write $\\delta-1$ as a product of factors bounded by the coordinate field sizes, and a recursive algorithm computes this count in $O(n/p^{r_m})$ operations. Tables of examples show enlarged codes whose parameters match or exceed the finite Gilbert-Varshamov bound for stabilizer quantum codes, and two of the codes meet the quantum Singleton bound, hence are optimal.","feed_headline":"Quantum codes gain K extra dimensions from Steane enlargement","feed_subtitle":"An arithmetic factorisation count predicts the exact dimension jump; several codes beat the Gilbert–Varshamov bound.","key_machinery":"The load-bearing object is the $\\sigma$-map on monomials, $\\sigma(X^a)=\\prod_{j=1}^m (p^{r_j}-a_j)$, which orders the exponent set $\\Delta(r)$ by designed distance: $C(L(\\delta))$, the span of evaluations of monomials with $\\sigma\\ge\\delta$, has true distance $\\delta$. Steane enlargement works because the nested code $C(L(\\delta-1))$ differs from $C(L(\\delta))$ exactly on the monomials with $\\sigma=\\delta-1$, and the number of such monomials is the arithmetic count $\\tau^{(r)}(\\delta-1)$ of factorisations $\\delta-1=\\prod_i d_i$ with $1\\le d_i\\le p^{r_i}$. This count is what Proposition 5 lower-bounds by $K$ via the edge values $1,2,\\dots,p^{r_i}$, what Corollary 9 refines by factoring $\\delta-1$, and what Algorithm 1 computes recursively by divisibility. The code pair's relative distances collapse to the usual minimal distances $d(C)=\\delta$ and $d(C')=\\delta-1$, so the enlargement theorem applies with the designed distance still at least $\\delta$.","core_discovery":"The central claim is Proposition 5: in the family of improved codes $C(L(\\delta))$ obtained by evaluating monomials whose $\\sigma$-value is at least $\\delta$, if $C(L(\\delta))$ is self-orthogonal and $2<\\delta\\le p^{r_2}+1$, then the pair $C(L(\\delta))\\subset C(L(\\delta-1))$ satisfies the hypotheses of the $q$-ary Steane-enlargement theorem. The enlargement therefore yields a quantum code with parameters $[[n, \\ge 2k-n+K, \\ge \\delta]]_q$, where $K$ is the largest index with $\\delta-1\\le p^{r_K}$. The proof exploits the edges of the exponent set, where $\\sigma$-values run through $1,2,\\dots,p^{r_i}$. Corollary 9 sharpens the dimension statement: the true gain is the number $\\tau^{(r)}(\\delta-1)$ of tuples $(d_1,\\dots,d_m)$ with $1\\le d_i\\le p^{r_i}$ and $\\prod_i d_i=\\delta-1$, so the guaranteed bound is exact exactly when $\\delta-1$ is prime; otherwise the gain is at least $K+\\binom{K}{2}$ for square $\\delta-1$ and at least $K^2$ otherwise. Algorithm 1 computes $\\tau^{(r)}$ recursively in $O(n/p^{r_m})$ operations, giving the true enlarged dimension in the examples.","pith_inferences":["Inference: the same $\\tau$-counting argument should transfer to any code family whose distance function factors over coordinates as a product of bounded ranges, such as other monomial-Cartesian or hyperbolic codes, whenever the dual description and self-orthogonality condition hold.","Inference: because the exact gain depends only on the arithmetic of $\\delta-1$ and the field sizes, Steane enlargement here is insensitive to finer geometry of the point set; a testable prediction is that codes from different Cartesian products with the same $r$-vector and same $\\delta$ have identical enlargement gains.","Inference: the examples suggest that small-distance enlargements outperform comparison constructions while larger-distance enlargements reverse the trend; systematically locating the threshold where the comparison flips would be a natural next step.","Inference: for $\\delta>p^{r_2}+1$, Proposition 5 cannot guarantee enlargement, yet the examples show that enlargement still succeeds in some cases, so a full characterisation of the success region is left open."],"forward_implications":["Every self-orthogonal $C(L(\\delta))$ with $2<\\delta\\le p^{r_2}+1$ can be Steane-enlarged to a quantum code with dimension at least $2k-n+K$ and distance at least $\\delta$, so the designed distance is never reduced.","The dimension increase is exactly $\\tau^{(r)}(\\delta-1)$; in particular it is exactly $K$ when $\\delta-1$ is prime, and at least $K+\\binom{K}{2}$ or $K^2$ when $\\delta-1$ is square or neither.","Algorithm 1 computes the true dimension increase in $O(n/p^{r_m})$ operations, which is a factor $p^{r_m}$ faster than enumerating all $n$ monomials.","Several of the enlarged codes match or exceed the finite Gilbert-Varshamov bound for stabilizer quantum codes, and the codes $[[27,23,3]]_9$ and $[[64,60,3]]_8$ meet the quantum Singleton bound, hence are optimal."],"supporting_citations":[{"why":"Supplies the classical codes $C(L(\\delta))$ from Cartesian product point sets, their parameters, and the self-orthogonality properties this paper assumes.","marker":"[4]"},{"why":"Gives the $q$-ary Steane-enlargement theorem used to produce the quantum codes.","marker":"[12]"},{"why":"Introduces the original Steane enlargement idea that the paper adapts to this setting.","marker":"[17]"},{"why":"Provides the finite Gilbert-Varshamov bound used to benchmark the enlarged codes.","marker":"[3]"},{"why":"Supplies the criterion for when two evaluated monomials are non-orthogonal, used in the proof of the dual code description.","marker":"[5]"},{"why":"Provides the footprint bound used to establish the minimal-distance statement for $C(L(\\delta))$.","marker":"[6]"},{"why":"Introduces the $\\tau$-map whose generalisation to arbitrary $r$-vectors underlies the exact dimension count and Algorithm 1.","marker":"[2]"},{"why":"States the quantum Singleton bound used to identify the two optimal codes.","marker":"[15]"}],"fun_headline_variants":["Steane enlargement gains exact quantum code dimensions via divisor counts","Divisor-count formula pins down Steane enlargement dimension boost","Steane-enlarged quantum codes beat Gilbert-Varshamov with exact gains","Arithmetic count gives exact dimension increase in Steane enlargement"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction rests on the unproved assumption, carried over from prior work, that each listed $C(L(\\delta))$ is self-orthogonal with the quoted $[n,k,d]$ parameters; if that fails for a row, the enlarged quantum code does not exist as claimed.","fun_headline_variants_meta":{"raw":{"variants":["Steane enlargement gains exact quantum code dimensions via divisor counts","Divisor-count formula pins down Steane enlargement dimension boost","Steane-enlarged quantum codes beat Gilbert-Varshamov with exact gains","Arithmetic count gives exact dimension increase in Steane enlargement"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000505,"raw_usage":{"total_tokens":2473,"prompt_tokens":961,"completion_tokens":1512,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":577,"completion_tokens_details":{"reasoning_tokens":1440}},"tokens_in":577,"tokens_out":1512,"duration_ms":12225,"temperature":1.0,"reasoning_tokens":1440,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:40:27.441399+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For any row of the tables, compute the Euclidean dual $C(L(\\delta))^\\perp$ and check whether it is contained in $C(L(\\delta))$; the claimed enlarged quantum code exists only if containment holds. Alternatively, enumerate all exponent tuples with $\\sigma$-value $\\delta-1$ and compare the count with the output of Algorithm 1.","supporting_citations":[{"cited_title":"IEEE Trans","cited_arxiv_id":null,"evidence_quote":"Supplies the classical codes $C(L(\\delta))$ from Cartesian product point sets, their parameters, and the self-orthogonality properties this paper assumes."},{"cited_title":"IEEE Trans","cited_arxiv_id":null,"evidence_quote":"Gives the $q$-ary Steane-enlargement theorem used to produce the quantum codes."},{"cited_title":"IEEE Trans","cited_arxiv_id":null,"evidence_quote":"Introduces the original Steane enlargement idea that the paper adapts to this setting."},{"cited_title":"Quantum Inf","cited_arxiv_id":null,"evidence_quote":"Supplies the criterion for when two evaluated monomials are non-orthogonal, used in the proof of the dual code description."},{"cited_title":"Steane-Enlargement of Quantum Codes from the Hermitian Curve","cited_arxiv_id":"1904.10007","evidence_quote":"Introduces the $\\tau$-map whose generalisation to arbitrary $r$-vectors underlies the exact dimension count and Algorithm 1."}],"review_version":1}